IMHO, excessive ‘perfectionism’ is also a by-product of Narcissism, as it’s really a desire for maintaining ‘control’ over one’s world, including everyone and everything in it…. according to our standards.
And just as with perfectionism, Narcissism begins very early, when our first sense of “self’ is being developed. But also as you say, over-demanding parents can negatively influence that process, to the point where we may lose connection and awareness of the many other parts of our personality. In which case, the key to reducing both Narcissism and ‘Perfectionism’, is to try and get back in touch with our innermost, and truest Self.